A survey commissioned by the trust found that more than a quarter of middle-class parents say they have moved house to live in areas with good schools, with more than 15% of parents from upper income bands saying they had moved to be near a specific school.

A minority of parents with children at state schools admitted to fiddling the admissions system to get into the school of their choice, with 2% of parents admitting they bought a second home and used that address so their children could gain access to a specific school, including 5% of those in the upper-middle classes.

"School admissions need to be fairer so that the best schools aren't just for those who can afford to live nearby, with ballots used particularly in urban areas," said Sir Peter Lampl, chairman of the Sutton Trust. "The government should consider extending its pupil premium to provide means-tested vouchers to enable working class parents to provide the extra lessons and cultural activities that many better-off families take for granted."

The survey conducted by YouGov of more than 1,000 parents in England found 3% admitting that they used a relative's address to get a child into a particular school, a figure that rose to 6% for members of the upper-middle classes.

Meanwhile, 6% of parents admitted attending church services, when they had not previously, to allow their child to go to a church school – including 10% in the upper-middle classes. But the report – written by two professors of education, Becky Francis and Merryn Hutchings – found little difference between classes when it came to cost-free methods of accessing good schools, disproving the idea that the middle classes are unusually enthusiastic about their children's education.

While 20% of parents from upper income bands said they had employed a tutor to provide help for their child to pass an entrance exam – a group that includes applications to independent and selective state schools – fewer than 5% of other parents did the same.

But all classes were enthusiastic in using cost-free alternatives, such as attending church services and appealing against unfavourable admissions decisions.

"Our research shows just how far equality of opportunity is being undermined by the greater purchasing power of some parents.

"The ability for some parents but not others to use financial resources to secure their children's achievement poses real impediments for social mobility, which need to be recognised and addressed as detrimental to society," said Prof Francis, one of the report's authors.

"However, our findings also demonstrate the extent to which some working-class parents are enacting informed choice, and policymakers may also learn from their practices."

The research found that level of parental education was the best predictor of how many sources of information parents used to research schools, with the highly educated most likely to use multiple sources of information. But a significant portion of all income classes – ranging from 10% for the upper-middle class to 40% for those on benefits or out of work – used one or no sources of information before choosing a school.

The research found that parents were more likely to make their decision using school open days and by talking to other parents rather than reading Ofsted reports and school prospectuses or school performance tables.